Juan D Lejarreta

SO(2,1)-invariant systems and the Berry phase

The general quadratic time-dependent quantum Hamiltonian is analysed using a time-dependent realisation of its SO(2,1) invariance Lie algebra. Several interesting features of this procedure are pointed out. As an example, the authors easily calculate dynamical and geometrical (Berry) phases using only an algebraic procedure. An incorrect result previously published by another author is also pointed out.

The generalized harmonic oscillator and the infinite square well with a moving boundary

For the one-dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in (i) the algebraic properties of the dynamical symmetry of the system and (ii) the construction of an invariant operator. The total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion. Two possible reductions for the system exist: the static harmonic oscillator and the free particle. In the latter case the quantum system becomes a Fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules. The time-dependent boundary conditions in the well play the role of an effective interaction acting on the particle. The formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed.

Non-isospectral 1+1 hierarchies arising from a Camassa--Holm hierarchy in 2+1 dimensions

The non-isospectral problem (Lax pair) associated with a hierarchy in 2 + 1 dimensions that generalizes the well known Camassa–Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose 1 + 1 Lax pair is also non-isospectral.

The Floquet analysis and noninteger higher harmonics generation

We consider here the exact solution of a nonrelativistic quantum system composed of a two-level atom interacting with a laser with arbitrary large frequency and intensity. We use the analogy of this system and a ℏ/2-spin particle interacting with a time-dependent magnetic field. A systematic use of the dynamical symmetry underlying the physical system is made. Actually the Hamiltonian is a Hermitian element of the SU(2) Lie Algebra. The exact Temporal Evolution Operator in terms of a Generalized Displacement Operator of the group is constructed. It is possible to develop a nonperturbative method that allows us to solve exactly the model for any value of the relevant frequencies (Rabi, Laser, and Atom Frequencies) and in so doing the usual Rotating Wave and Small Interaction approximations are unnecessary. The properties of periodicity of this model and the phenomenon of harmonic generation are considered by using Floquet Analysis. We find that in addition to the so far well-known spectrum composed by odd ...

An exact transmission coefficient with one and two barriers in graphene

In this work, we find the exact transmission coefficient across one and two potential barriers in graphene. We obtain theoretical expressions for this coefficient in terms of the characteristics of the potential barriers, height, width and separation and the characteristics of the carriers, Fermi energy and incidence angle. This coefficient is essential in order to characterize the conductance in several electronic devices based on graphene. Since the calculated transmission coefficient is highly non-isotropic, we extend the Landauer formalism and obtain a closed expression for transistor conductance.

Integrable 1+1 dimensional hierarchies arising from reduction of a non-isospectral problem in 2+1 dimensions

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in

Dynamical symmetries, cyclic and periodic states

2+1

Quantum phases and the degenerate optical parametric oscillator

dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in

Quantized Electron Transport Through Graphene Nanoconstrictions

2+1

The symmetry group of the quantum harmonic oscillator in an electric field

dimensions. A classification of reductions for this spectral problem is performed. Non-isospectral reductions in